1 矩阵\(Y=f(x)\)对标量x求导
矩阵Y是一个\(m\times n\)的矩阵，对标量x求导，相当于矩阵中每个元素对x求导
\[\frac{dY}{dx}=\begin{bmatrix}\dfrac{df_{11}(x)}{dx} & \ldots & \dfrac{df_{1n}(x)}{dx} \\ \vdots & \ddots &\vdots \\ \dfrac{df_{m1}(x)}{dx} & \ldots & \dfrac{df_{mn}(x)}{dx} \end{bmatrix}\]
2 标量y=f(x)对矩阵X求导
注意与上面不同，这次括号内是求偏导，\(X\)是是一个\(m\times n\)的矩阵，函数\(y=f(x)\)对矩阵\(X\)中的每个元素求偏导，对\(m\times n\)矩阵求导后还是\(m\times n\)矩阵
\[\frac{dy}{dX} = \begin{bmatrix}\dfrac{\partial f}{\partial x_{11}} & \ldots & \dfrac{\partial f}{\partial x_{1n}}\\ \vdots & \ddots & \vdots \\\dfrac{\partial f}{\partial x_{m1}} & \ldots & \dfrac{\partial f}{\partial x_{mn}}\end{bmatrix}\]
3 函数矩阵Y对矩阵X求导
矩阵\(Y=F(x)\)对每一个\(X\)的元素求导，构成一个超级矩阵
\[F(x)=\begin{bmatrix}f_{11}(x) & \ldots & f_{1n}(x)\\ \vdots & \ddots &\vdots \\ f_{m1}(x) & \ldots & f_{mn}(x) \end{bmatrix}\]
\[X=\begin{bmatrix}x_{11} & \ldots & x_{1s}\\ \vdots & \ddots &\vdots \\ x_{r1} & \ldots & x_{rs}\end{bmatrix}\]
，其中
\[\frac{dF}{dX} = \begin{bmatrix}\dfrac{\partial F}{\partial x_{11}} & \ldots & \dfrac{\partial F}{\partial x_{1s}}\\ \vdots & \ddots & \vdots \\\dfrac{\partial F}{\partial x_{r1}} & \ldots & \dfrac{\partial F}{\partial x_{rs}}\end{bmatrix}\]
其中
\[\frac{\partial F}{\partial x_{ij}} = \begin{bmatrix}\dfrac{\partial f_{11}}{\partial x_{ij}} & \ldots & \dfrac{\partial f_{1n}}{\partial x_{ij}}\\ \vdots & \ddots & \vdots \\\dfrac{\partial f_{m1}}{\partial x_{ij}} & \ldots & \dfrac{\partial f_{mn}}{\partial x_{ij}}\end{bmatrix}\]
\[\frac{\partial F}{\partial x_{ij}} = \begin{bmatrix} \end{bmatrix}\]
重要结论：假设
是一个向量：
，
，